3 weeks, at which point the whole city would be infected.
(Edited: half the city would be infected or dead and investigations would start about a week and a few days from the initial Patient Zero infection.)
Suppose Patient Zero acquires these symptoms and they show up in 48 hours; they then begin to spread this around. If everyone interacts with, say, 30 people a day and sneezes while talking to 25 of them, perhaps 15 of them will get infected from the spores depending on how infectious it is. Since Patient Zero isn’t taking any precautions since this is a new disease, it is possible that when they sneeze and throw out the tissue the spores remain there and infect whoever goes near that garbage bin for the rest of the day (say, 5).
In any case, say 20 people total are infected by Patient Zero on this third day. These twenty people, in turn, then each infect 20 people two days later with Patient Zero infecting 20 more people each day.
This would spread rapidly through the population depending on how large the city is, and even by the second or third week the whole capital would be infected - this because perhaps Patient Zero infects someone who meets many many people a day or someone who isn’t hygienic and sneezes on their hands and smears it all over public transport… the opportunities for infection are endless. Perhaps by this point - the second week when about ⅔ the population is infected - an investigation would begin, but it probably wouldn’t be fully found out by week 3 when everyone’s infected. Then the whole city would go into quarantine and if anyone had travelled in those 3 weeks they could spread it across the world, and panic would ensue.
This could be even faster if it can infect rats and other non-animals; they could, in turn, spread it from the city.
It could be slower if someone gets worried and goes to healthcare services; however, the symptoms are benign enough no one would pay attention.
Without further details about the nature of the spores and how long they can last on inorganic material and how big the city is, it would be hard to say.
EDITED to include new information:
If deaths occur within five days to a week, then by a week and a half investigations would surely start as people start dropping dead like flies. But probably about half of the people would be dead by the time they figure out what's going on and more dying by the moment while they race to quarantine people